Linear Functions Lecture Notes

Vocabulary flashcards covering the definitions, representations, and properties of linear functions based on the lecture transcript.

Linear Function

A function defined by $y = mx + b$ or $f(x) = mx + b$, where $m$ and $b$ are real numbers.

Slope ($m$)

The steepness of a line, represented by $m$ in the slope-intercept form.

$y$-intercept ($b$)

The point where the line crosses the $y$-axis, representing the value of $y$ when $x = 0$.

$$f(x)$$

A notation read as "$f$ of $x$" or "$f$ at $x$".

Degree

The highest exponent of the variable in an equation; for a linear function, this may be $0$, $1$, or undefined.

Table of Values Linearity

To determine if a table represents a linear function, the first differences of the $x$-coordinates must be equal and the first differences of the $y$-coordinates must be equal.

Independent Variable

The variable in a relationship that is controlled or changed.

Dependent Variable

The variable that depends on the independent variable in a linear relationship.

$x$-axis

The horizontal number line on a coordinate plane.

$y$-axis

The vertical number line on a coordinate plane.

Positive Slope Slant

When $m$ is positive, the graph slants upward from left to right, indicating the line is increasing.

Negative Slope Slant

When $m$ is negative, the graph slants downward from left to right, indicating the line is decreasing.

Horizontal Line

The result when the slope ($m$) is zero, appearing as a flat line across the coordinate plane.

Constant Function

A function that takes the same value for $f(x)$ no matter what $x$ is, typically in the form $f(x) = b$ with a slope of $0$.

Slope Formula

The formula used to find the slope between two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Slope ($m$) using Table of Values

The rate of change calculated as the change in $y$ divided by the change in $x$, or $\frac{\Delta y}{\Delta x}$.